The present invention relates to determining atomic structure using x-ray techniques. More specifically, the invention relates to an x-ray method for the determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal.
Many systems of current interest, both from a scientific and technological standpoint, can be characterized as two-dimensional crystals. Such systems are fully or partially periodic in two dimensions, a-periodic in the third, and positionally correlated with an underlying substrate crystal. Examples include reconstructed crystal surfaces, layered heterostructures, crystalline-amorphous interfaces (e.g., Sixe2x80x94SiO2), and self-assembled systems crystallized on a substrate. These systems are interesting because their physical properties differ markedly from those of bulk materials. Their study has been hampered by a lack of characterization tools that can probe these types of systems with atomic resolution.
Various x-ray methods have been used to investigate 2D structures. These include reflectivity and diffuse scattering (Sinha S K, et.al. 1994 X-ray scattering studies of surface roughness of Ga As/AlAs multilayers Physica B 198 72-7; and Miceli P F 1993 X-ray reflectivity from heteroepitaxial layers Semiconductor Interfaces, Microstructures and Devices: Properties and Applications (Bristol: Institute of Physics Publishing) pp 87-114), high-resolution x-ray diffraction along Bragg and truncation rods (Robinson I K and Tweet D J 1992 Surface X-ray diffraction Rep. Prog. Phys 55 599-651), multiple diffraction (Morelhao S L and Cardoso L P 1993 Structural properties of heteroepitaxial systems using hybrid multiple diffraction in Renninger scans J. Appl. Phys. 73 4218-26), and standing waves (Zegenhagen J 1993 Surface structure determination with X-ray standing waves Surf. Sci. Rep. 18 7-8). In all these methods, it is necessary to first postulate a structural model and then refine its parameters to obtain the best fit with experiment. Thus, if the model is incorrect, the structure will necessarily be incorrect.
Interface structures are often quite complex and guessing a reasonable structural model can be both difficult and unreliable. Two direct methods, proposed in the past, are anomalous reflectivity (Sanyal M K, et.al. 1993 Fourier reconstruction of density profiles of thin films using anomalous X-ray reflectivity Europhys. Lett.21 691-6) and x-ray holography (Tegze M and Faigel G 1996 X-ray holography with atomic resolution Nature 380 49-51). These methods are of limited use for 2D structures. The former only provides in-plane-averaged results. The latter provides the average structure around probe atoms that are typically located at in equivalent sites. In special cases when the system has inversion symmetry, the problem is greatly simplified because the scattering factors are real. This has been utilized to obtain the scattering factors of Fe/Ru superlattices and to calculate their structure by Fourier back-transformation (De S M, De A A, Raoux D, Maurer M, Ravet M F and Piecuch M 1992 Anomalous X-ray diffraction of a hexagonal Fe/Ru superlattice Phys. Rev. B 46 15 465-71).
Recently, using the tangent formula of Rius et al. (Rius J, Miravitlles C and Allman R 1996 A tangent formula derived from Patterson-function arguments. IV. The solution of difference structures directly from superstructure reflections Acta Crystallogr. A 52 634-9 ), Torrelles et al. (Torrelles X, et. al. 1998 Application of X-ray direct methods to surface reconstructions: the solution of projected superstructures Phys. Rev. B57 R 4281-4) developed a method for obtaining directly the projection of a reconstructed crystal surface on the surface plane from the Bragg-rod diffraction intensities on the equatorial plane in reciprocal space. The method has been generalized (Torrelles, et. al. 1999 Application of the xe2x80x9cdirect methodsxe2x80x9d difference sum function to the solution of reconstructed surfaces Surf. Sci. 423 2-3) to obtain the three-dimensional structure using the intensities on additional points along the Bragg rods. The problem of this method is that it involves the refinement of a large number of phase angles. Thus, the computation complexity grows at least as the square of the number of atoms.
It is therefore an object of this invention to provide a method for the determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal that is much simpler than the methods of the prior art.
It is another object of this invention to provide a method for direct determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal that does not require an a priori correct conjecture of the structure.
It is yet another object of this invention to provide a method for direct determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal that is general and does not depend on the symmetry properties of the system.
It is a further object of this invention to provide a method for direct determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal that provides the three dimensional structure of the system.
It is yet an additional object of this invention to provide a method for direct determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal whose computational complexity scales linearly with the number of atoms.
Further objects and advantages of this invention will appear as the description proceeds.
The present invention is directed to a general method for determining the three-dimensional structure of systems that are periodic in two dimensions, a-periodic in the third, and are commensurate with an underlying three-dimensional crystal. The system is considered as composed of two components: an xe2x80x9cunknown systemxe2x80x9d periodic in two dimensions, a-periodic in the third with an unknown structure and a xe2x80x9cknown systemxe2x80x9d also periodic in two dimensions with a known structure. The two systems are commensurate with each other. The sum of their electron densities is equal to the electron density of the entire system. The method provides the structure of the unknown and therefore of the entire system. The structure the method provides does not depend on an a priori correct model of the structure, can handle systems with large 2D unit cells and large layer thickness, and is non-destructive. Using diffraction data along one Bragg rod, the system layer structure can be reconstructed with monolayer resolution; and the full three-dimensional structure of a system can be reconstructed with atomic resolution from diffraction data along the Bragg rods within a certain volume in reciprocal space.
In a first aspect, the present invention is directed towards providing an x-ray method for the determination of the structure of systems that have two-dimensional periodicity and are positionally correlated with an underlying substrate crystal comprising the following steps:
creating two overlapping x-ray beams in the system having two-dimensional periodicity such that the overlapping beams propagate at an angle with respect to each other and their orientation relative to the two-dimensional system is such that the diffracted beams interfere with each other;
Measuring the diffraction intensity and diffraction interference patterns;
determining the phase derivative of the total complex scattering factor (CSF) along the Bragg rods using the diffraction interference patterns;
determining the CSF along the Bragg rods from the measured diffraction intensity, the phase derivative of the total CSF, and the CSF of the known system;
determining the electron-density function of the layer structure of the system by Fourier transforming the CSF along one Bragg rod;
performing a three-dimensional Fourier back-Transform into real space to provide the three-dimensional real space x-ray dielectric function; and
determining the three-dimensional spatial structure, of the system having two-dimensional periodicity from the three-dimensional x-ray dielectric function.
In the method of the experiment, the phase derivative of the total CSF is measured by employing the two-beam diffraction interference method (Yacoby Y 1994 Structure Factor amplitude and phase determination by a new two beam diffraction interference method Solid State Commun. 91 529-33 and Baltes H, Yacoby Y, et.al. 1997 Measurement of the X-ray diffraction phase in a 2D crystal Phys. Rev. Lett. 79 1285-8). The phase is then determined by either direct integration or preferably by an iterative method using the measured diffraction amplitudes, the phase derivative of the CSF, and the CSF of the known system.
In another aspect, the present invention is directed towards providing an x-ray method for the determination of the structure of systems that have two-dimensional periodicity, are positionally correlated with an underlying substrate crystal, and wherein the CSF of the unknown system varies slowly along the Bragg rods compared to that of the known system comprising the following steps:
Measuring the diffraction intensity along the Bragg rods;
locating the zero point of the real space z-coordinate such that changes in the CSF of the unknown system for two adjacent points along a Bragg rod are negligible compared to the changes in the CSF of the known system for the corresponding points;
determining the CSF along the Bragg rods from the measured diffraction intensity patterns and the CSF of the known system;
determining the electron-density function of the layer structure of the system by Fourier transforming the CSF along one Bragg rod;
performing a three-dimensional Fourier back-Transform into real space to provide the three-dimensional real space x-ray dielectric function; and
determining the three-dimensional spatial structure, of the system having two-dimensional periodicity from the three-dimensional x-ray dielectric function.
In this aspect of the invention, wherein, in addition, the thickness of the unknown structure having two-dimensional periodicity is small compared to its distance from the system surface, then the zero point of the real space z-coordinate is preferably placed within the unknown two-dimensional layer.
In a further aspect, the invention is directed towards providing an x-ray method for the determination of the periodic component of the structure of systems that have partial two-dimensional periodicity, are positionally correlated with an underlying substrate crystal, comprising the following steps:
creating two overlapping x-ray beams in the system having two-dimensional periodicity such that the overlapping beams propagate at an angle with respect to each other and their orientation relative to the two-dimensional system is such that the diffracted beams interfere with each other;
Measuring the diffraction intensity and diffraction interference patterns;
determining the phase derivative of the total complex scattering factor (CSF) along the Bragg rods using the diffraction interference patterns;
determining the CSF along the Bragg rods from the measured diffraction intensity, the phase derivative of the total CSF, and the CSF of the known system;
determining the electron-density function of the layer structure of the system by Fourier transforming the CSF along one Bragg rod;
performing a three-dimensional Fourier back-Transform into real space to provide the three-dimensional real space x-ray dielectric function; and
determining the three-dimensional spatial structure, of the system having two-dimensional periodicity from the three-dimensional x-ray dielectric function.
In yet a further aspect, the invention is directed towards providing an x-ray method for the determination of the periodic component of the structure of systems that have partial two-dimensional periodicity, are positionally correlated with an underlying substrate crystal, and wherein the CSF of the unknown system varies slowly along the Bragg rods compared to that of the known system comprising the following steps:
Measuring the diffraction intensity along the Bragg rods;
locating the zero point of the real space z-coordinate such that changes in the CSF of the unknown system for two adjacent points along a Bragg rod are negligible compared to the changes in the CSF of the known system for the corresponding points;
determining the CSF along the Bragg rods from the measured diffraction intensity patterns and the CSF of the known system;
determining the electron-density function of the layer structure of the system by Fourier transforming the CSF along one Bragg rod;
performing a three-dimensional Fourier back-Transform into real space to provide the three-dimensional real space x-ray dielectric function; and
determining the three-dimensional spatial structure, of the system having two-dimensional periodicity from the three-dimensional x-ray dielectric function.
In this aspect of the invention, wherein, in addition, the thickness of the unknown structure having partial two-dimensional periodicity is small compared to its distance from the system surface, then the zero point of the real space z-coordinate is preferably placed within the unknown two-dimensional layer.
According to the method of the invention, the two overlapping x-ray beams in the structure having two-dimensional periodicity, or partial two dimensional periodicity, are obtained by total external reflection from a heavy metal film on the sample surface.
All the above and other characteristics and advantages of the invention will be further understood through the following illustrative and non-limitative description of preferred embodiments thereof, with reference to the appended drawings.